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The contraction morphism between maps and quasimaps to toric varieties

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Pith's one-line read For every smooth projective toric variety, the paper constructs a contraction morphism from a closed substack of stable maps to the quasimap space, and proves it is surjective when the target is Fano.

desk verdict A genuinely new toric contraction morphism with a useful basepoint invariant; the main theorem is believable and the flaws are presentation-level, so it deserves a real referee. read the letter →

arxiv 2412.16295 v1 pith:CGAX2IZ4 submitted 2024-12-20 math.AG

Alberto Cobos Rabano This is my paper · ORCID
classification math.AG MSC 14N3514M2514D23
keywords toricvarietiesquasimapsstablemapscontractionmorphismbasepointdegreemoduliofcurvescurve-countinginvariantsFano
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper connects two compactifications of the space of maps from curves to a smooth projective toric variety: stable maps and stable quasimaps. For any smooth projective toric variety $X$ it constructs a closed substack of the stable-maps moduli space and a contraction morphism from that substack to the quasimap moduli space, agreeing with the identity on maps from smooth curves. When $X$ is Fano, the contraction morphism is proved to be surjective, so every stable quasimap is the contraction of an actual stable map. The new invariant that makes this work is the degree of a quasimap at a basepoint: an effective curve class attached to each basepoint that accounts exactly for the gap between the quasimap degree and the degree of its regular extension. This gives a concrete geometric mechanism for comparing the two enumerative invariants of the spaces.

What carries the argument

The central object is the degree of a basepoint (Definition 3.2.10): the unique effective curve class $\beta_x \in A_1(X)$ such that twisting the quasimap's line-bundle-section data by $-\beta_x$ at $x$ makes $x$ a regular point. This class is built combinatorially from the vanishing orders of the sections against maximal cones of the toric fan, and it satisfies the identity $\beta = \beta_{\mathrm{reg}} + \sum_{x\in B} \beta_x$. It supplies two mechanisms: it is the invariant that distinguishes quasimaps with the same regular extension, and it is the data used to graft a rational curve at a basepoint in the proof of surjectivity. The contraction morphism itself is assembled by embedding $X$ into a product of projective spaces through an epic closed embedding and restricting the projective-space contraction; on a stable map it contracts rational tails while twisting the remaining sections by the tail degrees.

What would settle it

Compare, on the nonreduced family $\operatorname{Spec} \mathbb{C}[\varepsilon]/\varepsilon^2$, the two fiber-product substacks obtained from the two epic embeddings of $\mathrm{Bl}_0\mathbb{P}^2$ written in Example 4.2.6; the paper proves coincidence only on closed points, so any difference in the stacks or in $c_X$ would show the construction is choice-dependent. Separately, run the grafting algorithm of Section 6.2 on the quasimap of Example 5.2.4: if at any stage the required sections $t_\rho$ on the grafted $\mathbb{P}^1$ cannot be chosen, surjectivity for Fano targets would fail.

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Extended reading notes

Core claim

The paper's central claim is that the contraction morphism, previously available for projective space, can be constructed for every smooth projective toric variety. One chooses an epic closed embedding of $X$ into a product of projective spaces, meaning an embedding whose induced map on curve classes is injective; Corollary 4.3.4 shows such embeddings make the quasimap space of $X$ a closed substack of the quasimap space of the product. Pulling back the product's contraction morphism through this closed embedding defines $c_X$ on a closed substack $M^c_{g,n}(X,\beta)$ of the stable maps stack (Construction 5.1.2). On points, $c_X$ contracts each rational tail of a stable map and twists the remaining line-bundle sections by the tail's degree; Proposition 5.2.1 characterizes which stable maps lie in the substack. Theorem 6.0.1 proves surjectivity for Fano $X$ by grafting rational curves onto the basepoints of any quasimap until a stable map is obtained. Along the way the paper proves a quasimap is determined by its regular extension, its basepoints, and the degree of each basepoint.

Load-bearing premise

Everything hangs on one auxiliary choice: an embedding of $X$ into a product of projective spaces; the paper proves the points of the resulting substack are independent of that choice, but not that the substack as a whole is, so the uniqueness of "the" contraction morphism is not yet established.

Editorial extensions

If this is right

  • For smooth Fano toric $X$, every stable quasimap of class $\beta$ is the image under $c_X$ of a stable map of class $\beta$.
  • A quasimap is uniquely determined by its regular extension, the set of its basepoints, and the degree of each basepoint (Corollary 3.4.2).
  • The degree of a basepoint explains the difference in degree: $\beta = \beta_{\mathrm{reg}} + \sum_{x\in B} \beta_x$, with $\beta_x=0$ exactly away from basepoints.
  • Closed embeddings between toric varieties that are injective on curve classes induce closed embeddings of quasimap spaces (Corollary 4.3.4).
  • If every toric boundary divisor of $X$ is numerically effective, the contraction morphism is defined on the entire stable maps space rather than only a closed substack.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the embedding-independence issue is resolved, the same construction would give a canonical contraction morphism whose pointwise formula could be used to compare virtual fundamental classes of the two moduli spaces without wall-crossing.
  • The surjectivity proof reads as a terminating combinatorial algorithm on the fan: resolve basepoints by successively grafting rational curves, with the Fano condition guaranteeing the required section choices exist; running it on non-Fano toric surfaces should locate precisely where the process can loop forever.
  • The degree of a basepoint is finer than the previously known length invariant, so it may distinguish quasimaps that have the same length; the two quasimaps in Example 4.1.1 are a natural test case.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper constructs, for every smooth projective toric variety X, a closed substack M^c_{g,n}(X,β) of the stable maps stack and a morphism c_X from it to the toric quasimaps stack Q_{g,n}(X,β), extending the identity on maps from smooth curves. The construction is based on a new invariant: the degree β_x of a quasimap at a basepoint, defined combinatorially and characterised by a universal twisting property. The paper proves that a quasimap is determined by its regular extension and the degrees at its basepoints, that pushforwards along epic closed embeddings of quasimap spaces are closed embeddings, and that for Fano X the contraction morphism is surjective. The motivation is a geometric comparison between Gromov–Witten and quasimap invariants for toric targets.

Significance. If correct, the paper provides a genuinely useful comparison morphism for all smooth projective toric varieties, generalising the contraction map for projective spaces and for one parametrized component. The basepoint-degree invariant is a valuable new tool: it explains the difference between quasimap degree and regular-map degree, recovers the length of a basepoint, and controls injectivity of functoriality maps between quasimap spaces. Theorem C, that Q(ι) is a closed embedding when ι is epic, is a clean structural result. The surjectivity theorem for Fano targets is a strong statement that could support future virtual-class comparison arguments. The main limitations are that the contraction construction is only proved to be independent of the auxiliary embedding at the level of closed points, and that the surjectivity proof contains a compressed induction step that needs to be made rigorous.

major comments (4)
  1. [Section 5.1, after Construction 5.1.2] Construction 5.1.2 defines M^c_{g,n}(X,β) and c_X using an auxiliary epic closed embedding ι into a product of projective spaces. The text acknowledges the a priori dependence on ι, but Proposition 5.2.1 only proves that the closed points of M^c and the pointwise formula for c_X are independent of ι. Since M^c is a closed substack of M_{g,n}(X,β), having the same closed points does not imply that the stack structures (nilpotent structure, universal family, or the closed immersion j) agree for different choices of ι. Thus the phrase 'the contraction morphism of X' is not yet justified as a canonical object. For each fixed ι the existence of a contraction morphism is not in question, but either a stack-level independence statement must be proved or the paper should systematically state the results for a chosen epic embedding.
  2. [Section 6.3, proof of Theorem 6.0.1] After reducing to one basepoint, the proof says 'by restricting ourselves to the irreducible component containing the basepoint x, we can assume that C is irreducible.' This reduction is not justified: restricting a quasimap to one irreducible component discards the other components, their nodes, and any markings on them, and a stable map constructed on that component does not automatically extend to a stable map on the original curve with c_X(f)=q. The multi-component case needs a separate argument — for example by grafting rational tails onto the remaining components and carefully tracking the total class, the markings, and stability — rather than a one-sentence reduction.
  3. [Section 1.1, definition of M^c] The introduction defines M^c_{g,n}(X,β) as Q_{g,n}(X,β) ×_{Q_{g,n}(P,ι_*β)} M_{g,n}(P,ι_*β). This is not the object constructed in Section 5.1, where the Cartesian square is taken with M_{g,n}(X,β) in the upper-left corner, i.e. M^c = M_{g,n}(X,β) ×_{Q_{g,n}(P,ι_*β)} Q_{g,n}(X,β). The introductory formula would not define a closed substack of M_{g,n}(X,β). Please correct the introductory definition so that it agrees with Construction 5.1.2.
  4. [Section 4.3, Corollary 4.3.4] Corollary 4.3.4 concludes that Q(ι) is a closed embedding because it is proper and a monomorphism. The proof of monomorphism cites Theorem 4.3.3, which explicitly describes fibres as sets of closed points. Since the statement concerns algebraic stacks over C, the argument should either spell out that the same description applies to all geometric points, or cite a criterion that makes closed-point injectivity plus properness sufficient for a monomorphism in this setting. As written, the step is slightly too compressed.
minor comments (5)
  1. [Section 3.2, after Definition 3.2.10] The sentence 'we came up with Definition 3.2.10 while studying...' is informal and out of place in a research paper; it should be removed or moved to an introductory remark.
  2. [Proof of Proposition 3.2.9] There is a typo: 'existance' should be 'existence'.
  3. [Remark 5.2.3] The word 'instrinsic' should be 'intrinsic'.
  4. [Diagram (24) and the Cartesian diagram in Construction 5.1.2] The typeset diagram is hard to parse. Adding explicit labels such as j, c_X, M(ι), cP, and Q(ι) on the actual arrows, or writing the fibre product in a displayed equation, would improve readability and prevent ambiguity.
  5. [Example 5.2.2] The phrase 'As a sanity check for Proposition 5.2.1' is informal; consider replacing it with 'As a verification of Proposition 5.2.1'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the contraction morphism is built from an independently defined basepoint-degree invariant and external projective-space contraction results, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is self-contained and does not reduce to its own inputs. The central new invariant, the degree βx of a basepoint, is defined directly from vanishing orders of the quasimap sections (Construction 3.2.3, Proposition 3.2.7, Proposition 3.2.9), not from the contraction morphism or from any fitted data. The identity β = βreg + Σ βx (Proposition 3.4.3(3)) is a consequence of the explicit twisting description of the regular extension in Corollary 3.4.1, so it is derived rather than assumed. The claimed recovery of the CFKM14 length (Lemma 3.5.1) is verified against an independent external notion, which is real supporting evidence rather than circularity. Theorem C / Corollary 4.3.4, giving closed embeddings of quasimap spaces for epic toric embeddings, is proved from the explicit fibre description Theorem 4.3.3, which itself follows from the degree-of-basepoint machinery; no part of that proof assumes the closed embedding conclusion. Construction 5.1.2 defines the contraction morphism as a fibre product using the independently established projective-space contraction morphism c_P and the closed embedding Q(ι); Proposition 5.2.1 then derives the pointwise description from this defining diagram rather than importing it. The Fano surjectivity theorem (Theorem 6.0.1) is proved by an induction on curve-class length with an explicit grafting construction (Construction 6.2.1), not by assuming surjectivity. The acknowledged dependence of Construction 5.1.2 on the auxiliary epic embedding ι, with Proposition 5.2.1 proving independence only on closed points, is a genuine canonicity/completeness gap but is explicitly disclosed by the paper and is not a circularity: it does not make the existence of the morphism or its surjectivity for each fixed ι equivalent to the input. No parameter is fitted to a subset of data and then renamed a prediction, and no load-bearing claim rests solely on a self-citation; references to the author's thesis are provenance, and the substantive cited results are established external constructions. The circularity score is therefore 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard toric geometry and quasimap theory plus one new mathematical invariant, the degree of a basepoint. No physical or geometric entities are postulated: beta_x is a defined invariant, not an explanatory entity, and the auxiliary epic embeddings are chosen, not fitted. There are no free parameters fitted to data.

assumptions (5)
  • standard math Standard facts on smooth projective toric varieties from CLS11: Cox ring presentation, Picard group basis from rays outside a maximal cone, nef and Mori cone duality, very ampleness of -K_X for smooth Fano toric varieties.
    Invoked in Sections 2.1, 3.2, 4.2, and 6.3; controls the existence of epic embeddings and the irreducibility and length arguments.
  • standard math Geometric invariant theory presentation of toric varieties and Cox's functor of points from Cox95.
    Basis for Definition 2.2.1, quasimap Definition 2.3.1, Construction 2.5.1, and the length comparison in Lemma 3.5.1.
  • standard math Properness, separatedness, and Deligne-Mumford stack structure of Q_{g,n}(X,beta) and M_{g,n}(X,beta) from CFK10 and the Stacks Project.
    Used in Theorem 2.4.4, Corollary 4.3.4, and the surjectivity criterion in Theorem 6.0.1.
  • standard math Existence and properties of the contraction morphism for projective spaces and products of projective spaces from MOP11 and PR03.
    The toric contraction morphism is built by embedding into a product of projective spaces and pulling back cP; without this input Construction 5.1.2 would have no starting point.
  • domain assumption The stability assumption 2g-2+n >= 0 for quasimap moduli spaces.
    Remark 2.4.2 rules out unstable cases; the paper assumes this inequality throughout.

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Pith. "Pith review of The contraction morphism between maps and quasimaps to toric varieties." pith.science (2026). https://pith.science/paper/CGAX2IZ4

@misc{pith2026241216295,
  author       = {Pith},
  title        = {Pith review of: The contraction morphism between maps and quasimaps to toric varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CGAX2IZ4}},
  note         = {Machine review of arXiv:2412.16295}
}
abstract

Given $X$ a smooth projective toric variety, we construct a morphism from a closed substack of the moduli space of stable maps to $X$ to the moduli space of quasimaps to $X$. If $X$ is Fano, we show that this morphism is surjective. The construction relies on the notion of degree of a quasimap at a base-point, which we define. We show that a quasimap is determined by its regular extension and the degree of each of its basepoints.

Figures

Figures reproduced from arXiv: 2412.16295 by the authors.

Figure 1
Figure 1. The fan of P 2 . Indeed, we can write each section si as si = z dx x s ′ i with zx a local coordinate at x and s ′ i ∈ H0 (C, L ⊗ OC(−dxx)). Furthermore, if the minimum in Equation (14) is achieved at si , then ord x(s ′ i ) = ord x(si) − dx = 0, so s ′ i (x) 6= 0. This ensures that x is not a basepoint of the quasimap (C, L′ , s′ 0 , . . . , s′ N ) with L ′ = L ⊗ OC(−dxx); thus proving the claim. Note that if we re… view at source ↗
Figure 2
Figure 2. The fan of Bl0P 2 . be a product of projective spaces. Quasimaps to P will be used in Section 5 to define the contraction morphism between stable maps and stable quasimaps. Given a quasimap q = (C, Lρ, sρ, cm), we can use the isomorphisms cm to view it as a collection of k line bundles L1, . . . , Lk on C together with a collection of ni + 1 sections s0,i, . . . , sni,i ∈ H0 (C, Li) for each i ∈ {1, . . . , k}. As i… view at source ↗

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Cited by 1 Pith paper

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  1. Irreducible components of moduli spaces of maps to smooth projective toric varieties in genus 0

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    The irreducible components of genus-0 stable map spaces to toric varieties are indexed by stable decorated trees that maximize a combinatorial invariant d_GGG.

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Works this paper leans on

26 extracted references · 20 canonical work pages · cited by 1 Pith paper

  1. [1]

    A L efschetz-type theorem for quasimap invariants via relative quasimaps

    Luca Battistella and Navid Nabijou. A L efschetz-type theorem for quasimap invariants via relative quasimaps. arXiv preprint arXiv:1710.11158v1 https://arxiv.org/abs/1710.11158v1, 2017

    arXiv 2017

  2. [2]

    Relative quasimaps and mirror formulae

    Luca Battistella and Navid Nabijou. Relative quasimaps and mirror formulae. International mathematics research notices , 2021(10):7885--7931, 2021

    work page 2021

  3. [3]

    Orbifold quasimap theory

    Daewoong Cheong, Ionu t Ciocan-Fontanine, and Bumsig Kim. Orbifold quasimap theory. Mathematische Annalen , 363:777--816, 2015

    work page 2015

  4. [4]

    Moduli stacks of stable toric quasimaps

    Ionuţ Ciocan-Fontanine and Bumsig Kim. Moduli stacks of stable toric quasimaps. Advances in mathematics (New York. 1965) , 225(6):3022--3051, 2010

    1965

  5. [5]

    Wall-crossing in genus zero quasimap theory and mirror maps

    Ionu t Ciocan-Fontanine and Bumsig Kim. Wall-crossing in genus zero quasimap theory and mirror maps. Algebr. Geom. , 1(4):400--448, 2014

    work page 2014

  6. [6]

    Big I -functions

    Ionu t Ciocan-Fontanine and Bumsig Kim. Big I -functions. In Development of moduli theory—Kyoto 2013 , volume 69, pages 323--348. Mathematical Society of Japan, 2016

    work page 2013

  7. [7]

    Higher genus quasimap wall-crossing for semipositive targets

    Ionu t Ciocan-Fontanine and Bumsig Kim. Higher genus quasimap wall-crossing for semipositive targets. Journal of the European Mathematical Society , 19(7):2051--2102, 2017

    work page 2017

  8. [8]

    Quasimap wall-crossings and mirror symmetry

    Ionu t Ciocan-Fontanine and Bumsig Kim. Quasimap wall-crossings and mirror symmetry. Publications math \'e matiques de l'IH \'E S , 131(1):201--260, 2020

    2020

Show all 26 references
  1. [9]

    Stable quasimaps to GIT quotients

    Ionuţ Ciocan-Fontanine, Bumsig Kim, and Davesh Maulik. Stable quasimaps to GIT quotients. Journal of geometry and physics , 75:17--47, 2014

    2014

  2. [10]

    Higher-genus wall-crossing in the gauged linear sigma model

    Emily Clader, Felix Janda, and Yongbin Ruan. Higher-genus wall-crossing in the gauged linear sigma model. Duke Math. J. , 170(4):697--773, 2021

    2021

  3. [11]

    Cox, John B

    David A. Cox, John B. Little, and Henry K. Schenck. Toric varieties , volume 124 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2011

    2011

  4. [12]

    The functor of a smooth toric variety

    David A Cox. The functor of a smooth toric variety. Tôhoku mathematical journal , 47(2):251--262, 1995

    1995

  5. [13]

    Gromov- W itten and quasimap invariants of toric varieties - a geometric perspective

    Alberto Cobos Rabano. Gromov- W itten and quasimap invariants of toric varieties - a geometric perspective. available at https://etheses.whiterose.ac.uk/35333/ https://etheses.whiterose.ac.uk/35333/, August 2024

    2024

  6. [14]

    The local/logarithmic correspondence and the degeneration formula for quasimaps

    Alberto Cobos Rabano, Cristina Manolache, and Qaasim Shafi. The local/logarithmic correspondence and the degeneration formula for quasimaps. arXiv preprint arXiv:2404.01381 https://arxiv.org/abs/2404.01381, 2024

    2024

  7. [15]

    Mirror symmetry for stable quotients invariants

    Yaim Cooper and Aleksey Zinger. Mirror symmetry for stable quotients invariants. Michigan Mathematical Journal , 63(3):571--621, 2014

    2014

  8. [16]

    A mathematical theory of the gauged linear sigma model

    Huijun Fan, Tyler Jarvis, and Yongbin Ruan. A mathematical theory of the gauged linear sigma model. Geom. Topol. , 22(1):235--303, 2018

    2018

  9. [17]

    A Mirror Theorem for Toric Complete Intersections , pages 141--175

    Alexander Givental. A Mirror Theorem for Toric Complete Intersections , pages 141--175. Birkh \"a user Boston, Boston, MA, 1998

    1998

  10. [18]

    Mirror theorem for elliptic quasimap invariants

    Bumsig Kim and Hyenho Lho. Mirror theorem for elliptic quasimap invariants. Geometry & Topology , 22(3):1459--1481, 2018

    2018

  11. [19]

    Derived moduli of sections and push-forwards

    David Kern, Etienne Mann, Cristina Manolache, and Renata Picciotto. Derived moduli of sections and push-forwards. arXiv preprint arXiv:2210.11386 https://arxiv.org/abs/2210.11386, 2022

    2022 arXiv

  12. [20]

    Stable quotients and the holomorphic anomaly equation

    Hyenho Lho and Rahul Pandharipande. Stable quotients and the holomorphic anomaly equation. Advances in Mathematics , 332:349--402, 2018

    2018

  13. [21]

    Virtual push-forwards

    Cristina Manolache. Virtual push-forwards. Geometry & Topology , 16(4):2003--2036, 2012

    2003

  14. [22]

    Stable maps and stable quotients

    Cristina Manolache. Stable maps and stable quotients. Compositio mathematica , 150(9):1457--1481, 2014

    2014

  15. [23]

    The moduli space of stable quotients

    Alina Marian, Dragos Oprea, and Rahul Pandharipande. The moduli space of stable quotients. Geometry and Topology , 15(3):1651--1706, 2011

    2011

  16. [24]

    Stable maps and quot schemes

    Mihnea Popa and Mike Roth. Stable maps and quot schemes. Inventiones mathematicae , 152(3):625--663, 2003

    2003

  17. [25]

    The stacks project

    The Stacks project authors . The stacks project. https://stacks.math.columbia.edu, 2022

    2022

  18. [26]

    Moduli spaces of stable quotients and wall-crossing phenomena

    Yukinobu Toda. Moduli spaces of stable quotients and wall-crossing phenomena. Compositio Mathematica , 147(5):1479--1518, 2011

    2011

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